Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > eqsupd | Structured version Visualization version GIF version |
Description: Sufficient condition for an element to be equal to the supremum. (Contributed by Mario Carneiro, 21-Apr-2015.) |
Ref | Expression |
---|---|
supmo.1 | ⊢ (𝜑 → 𝑅 Or 𝐴) |
eqsupd.2 | ⊢ (𝜑 → 𝐶 ∈ 𝐴) |
eqsupd.3 | ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ¬ 𝐶𝑅𝑦) |
eqsupd.4 | ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝐶)) → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧) |
Ref | Expression |
---|---|
eqsupd | ⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) = 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqsupd.2 | . 2 ⊢ (𝜑 → 𝐶 ∈ 𝐴) | |
2 | eqsupd.3 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ¬ 𝐶𝑅𝑦) | |
3 | 2 | ralrimiva 3180 | . 2 ⊢ (𝜑 → ∀𝑦 ∈ 𝐵 ¬ 𝐶𝑅𝑦) |
4 | eqsupd.4 | . . . 4 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝐶)) → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧) | |
5 | 4 | expr 459 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝑦𝑅𝐶 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)) |
6 | 5 | ralrimiva 3180 | . 2 ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 (𝑦𝑅𝐶 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)) |
7 | supmo.1 | . . 3 ⊢ (𝜑 → 𝑅 Or 𝐴) | |
8 | 7 | eqsup 8912 | . 2 ⊢ (𝜑 → ((𝐶 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝐶𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝐶 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)) → sup(𝐵, 𝐴, 𝑅) = 𝐶)) |
9 | 1, 3, 6, 8 | mp3and 1458 | 1 ⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1531 ∈ wcel 2108 ∀wral 3136 ∃wrex 3137 class class class wbr 5057 Or wor 5466 supcsup 8896 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-ral 3141 df-rex 3142 df-reu 3143 df-rmo 3144 df-rab 3145 df-v 3495 df-sbc 3771 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-nul 4290 df-if 4466 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4831 df-br 5058 df-po 5467 df-so 5468 df-iota 6307 df-riota 7106 df-sup 8898 |
This theorem is referenced by: supmax 8923 supiso 8931 dfgcd2 15886 esumpcvgval 31330 esum2d 31345 mblfinlem3 34923 mblfinlem4 34924 ismblfin 34925 itg2addnclem 34935 radcnvrat 40637 |
Copyright terms: Public domain | W3C validator |